A prior art nozzle piece of the kind as herein referred comprises in general an inner discharge nozzle for discharging a mixture of a gaseous carrier medium and an abrasive dental powder as supplied by an associated first supply duct of a gripping member which is connected at an end with the head portion of a handpiece. The nozzle piece further comprises a concentrically arranged outer discharge nozzle for discharging a fluid which is supplied by an associated second supply duct of the gripping member. Nozzle pieces of this kind are disclosed for example in U.S. Pat. Nos. 4,174,571; 4,412,402 and 4,676,749 in combination with a dental abrasive blasting apparatus which comprises a powder reservoir inside of a housing of the apparatus for storing the dental powder. The nozzle piece is fixed to the one end of two tubings which directly connect with inner and outer discharge nozzles of the nozzle piece and which at their opposite end are fixed on a head piece of a gripping sleeve of the handpiece. The two tubings extend in parallel with respect to a longitudinal axis of the handpiece and are running straight or under an angle and sometimes also along a curved path towards the discharge nozzles of the nozzle piece. By these different paths of the two tubings respective different discharge angles are provided for concentric discharge openings of the inner and outer discharge nozzles which have a common central axis with respect to the longitudinal axis of the handpiece. The discharge angle is usuangle is usually in the range of 60° or more. With the provision of a discharge angle of this range any visual contact with the exit of the working jet from the discharge openings of the two discharge nozzles at the tip of the handpiece is not at all taken into consideration and it is instead only intended to reach an optimum manipulation of the handpiece for securely guiding the working jet over the preparation field of the teeth in the course of an abrasive dental treatment. The same strategy may also be recognised for dental handpieces of the kind comprising an integrated powder reservoir as for example described in U.S. Pat. No. 4,776,794 and having also a nozzle piece with inner and outer discharge nozzles for discharging a working jet over a preparation field of the teeth.